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A MEASIBEMENT OP THE Ni^*^ DIE3CTI0NAL COHSELATION 
WITH A TUMT'RT. DIODE COUTGIDEHGE CIECUIT 






by 






GARY EDWIN CT.ARK 


« 




B. A., Park College, 1961 






A MASTER'S THESIS 






submitted in partial fulfillment of the 

* 






requirements for the degree 






MASTER OP SCIENCE 






Department of Physics 






KANSAS STATE UNIVERSITY 
Manhattan, Kansas. . 






1966 






Approved by; 






Major Professor 


■• 

n 



C91 TABLE OF CONTENTS 

I. INTRODUCTION 1 

1. Purpose of the Paper 1 

2. Review of Literature 2 

II. TUNNEL DIODE COINCIDENCE CIRCUIT ........ 3 

1. Properties of the Tunnel Diode 3 

2. The Uni vibrator Circuit 5 

5. Detection of Coincident Pulses 9 

4. Zero Crossover 10 

5. Noise and Feedthrough ......... 11 

6. Slewing 12 

7. Effect of Temperature lA- 

8. Jitter 14 

9. Nanosecond Systems, Inc., Equipment . • 14 

III. THEORY OP DIRECTIONAL CORRELATION 1? 

1. The Directional Correlation Function . . 17 

2. The Ni60 Cascade 20 

$. Analysis of Chance Rates 21 

4. Experimental Considerations 24 

5. Correction for Finite-Size Detectors . . 27 

IV. MEASUREMENT OF THE Ni^^ DIRECTIONAL 

CORRELATION 51 

1. Apparatus 51 

2. Determination of Resolving Times .... 52 
5. Determination of Chance Rates 55 

4. Determination of Correction for Finite- 

Size Detectors 54 

5. Reduction of Error at 155° ....... 55 

6. Analysis of Ni«^ Correlation Data ... 56 

V. CONCLUSION , 41 

VI. ACKNOWLEDGMENTS 42 

VII. PLATES. . . 45 

VIII . REFERENCES 75 



I. INTRODUCTION 

1. Purpose of the Paper 
In nuclear spectrometry scintillation materials and photo- 
multiplier tubes are frequently used which produce electrical 
pulses with two useful parameters, pulse height and time of 
occurrence. In this paper we are concerned with the second 
parameter and in particular a method of determining whether or 
not two pulses occur coincidently within a few nanoseconds 
(10"-' seconds) or less. Tunnel diodes, which are sometimes 
called Esaki diodes, are a suitable choice for a coincidence 
circuit for several reasons: They have fast switching times, 
can be very much overdriven without harmful effects, consume 
little power, and require only a small signal. Furthermore 
they are insensitive to humidity variations, can operate over 
a wide temperature range, and withstand neutron irradiation. 
The fact that they are two-terminal devices, and therefore not 
unidirectional, is a large disadvantage which can be somewhat 
overcome by careful circuit design. 

A txinnel diode coincidence circuit similar to the one 

48 
described by Whetstone was constructed and its properties 

are discussed in the second part of the paper. One application 

of coincidence circuits is in the measurement of a directional 

correlation, the theory of which is treated in Part III. The 

experimental determination of the directional correlation func- 

60 
tion of the Ni cascade using this circuit is given in Paj?t IV. 

Sufficient evidence is presented to indicate that the circuit 



can be used for this type of measurement. Before proceeding 
witb. the main part of this paper a short review of the litera- 
ture is presented. 

2. Review of Literature 

Scintillation methods have "been employed in nuclear 

51 
physics since the beginning, while coincidence techniques 

were used as early as 1929* Although most present day 
fast coincidence circuits utilize solid state devices, early 
circuits built by Rossi and others had vacuum tubes 
only. ^^'^^^ j'^^i'^o, i ^ ^g^ circuits are reported which com- 
bine tunnel diodes and vacuum tubes or tunnel diodes and 

1 55 46 
transistors. *^'^' Avalanche transistors have been employed 

2 15 

but not extensively. * ^ The complexity of coincidence 

circuits ranges from a simple circuit of crystal diodes to 
extremely complex combinations of electronic devices. 

There are many variations of the fairly simple and 
highly reliable circuit with which the paper is con- 
cerned. ^^»^^»^^'^5»^'''^ The basic building block (see 
Plate II) of such a circuit is the tunnel diode univibrator, 

which can handle coxinting rates in excess of 100 Mc/s pro- 

1 49 
vided dead time is properly controlled. * "^ Because tunnel 

diodes have extremely small switching times, this circuit is 

advantageous when a very small coincidence resolving time is 

desired. 

A word must be said about gallixim arsenide tunnel diodes, 

which were considered very unreliable a few years ago. Gallium 



arsenide tunnel diodes, when not subjected to continuous volt- 
age high above the forward point, and germanium tunnel diodes 
are now equally reliable and are available at reasonable 
prices. Their various properties will be discussed in the 
next section. 

II. TUMEL DIODE COINCIDENCE CIRCUIT 

1. Properties of the Tunnel Diode 
Simply stated, a tunnel diode is a heavily doped p-n 
semiconductor junction. A lightly doped junction has little 
resistance when forward biased and high resistance when reverse 
biased up to the reverse breakdown or Zener voltage. If the 
concentration of impurity atoms is increased, the reverse 
breakdown voltage approaches zero; and it might be thought 
that an infinite concentration of impurity atoms would be 
needed to attain a breakdown condition at zero applied voltage; 
however v/hen doping is increased beyond a definite concentra- 
tion, which is easily obtainable in practice, a reverse break- 
down condition exists even at a small forward bias, and the 
device is a tunnel diode. Such a diode exhibits negative 
resistance at a small forward bias. 

The static characteristic curve of a tunnel diode is 
shown in Plate I. The negative resistance region is between 
the peak point, labeled b, and the valley point, labeled d; 
and point c is called the forward point. Theoretically this 
cxirve should be the sum of a Gaussian and an exponential, 



"but it differs in two ways. First, the current in the valley 
region is always greater than would be expected, the differ- 
ence "being due to defect states in the forbidden gap energy 

o 

band as postulated by Yajima and Esaki. Second, the peak 
region is always skewed slightly to the right, this being 
attributed to stray circuit parameters which can never be 
completely eliminated. 

In order to make circuit calculations, the characteristic 

29 ^1^1 
curve may be approximated by two psirabolas * or a series 

22 

of straight lines. In the latter approximation the resist- 
ance in the low voltage region is given by the formula 

S,, = 0.75 "V /I » where Y^ is the peak voltage and I is the 
dl PP P P 

peak cvirrent; and the resistance in the high voltage region is 

given by R^2 ' ^^fp~ ^v^'^^^n" "^v^ ' where V^ is the forward 
point voltage, I is the valley current, and ■\r^ « 0.5(V^ - Y^) 
with "V being the valley voltage. The peak, valley, and for- 
ward point voltages, which are determined by the semiconductor 
material, are 55 mV, 359 21V, and 500 mV respectively at 25° C 

for germanium and about double these values for gallium 

21 
arsenide tunnel diodes. Measured values of parameters 

usually agree with those calculated from the linesirized, curve 
to within ten per cent. 

The peak current is determined by the impurity concentra- 
tion and the cross-sectional area of the junction. In prac- 
tice the junction is made larger than necessary and then 
etched away until the desired current is achieved. A rather 



high current density is used to decrease tlie capacitance, which 

depends on the peak current. A one milliampere unit, for 

—4- 
example, might have a junction diameter of less than 5 X 10 

inch. The ratio of peak current to junction capacitance, I^/C, 
is approximately constant for a given semiconductor material 
and doping, and is about 1/2 for the 1N5118 gallium arsenide 
and about 5 for the germanium tvmnel diodes utilized in our 
circuit. Because an input pulse must supply charge to the 
junction capacitance, and because the rate of charge accumula- 
tion is a function of overdrive, there is a time delay in the 
firing of the tunnel diode. The variation in this time delay 
with overdrive is called the slewing characteristic. The time 
delay at the input stage is reduced considerably by using a 
germanium tunnel diode, for which the junction capacitance is 
about ten times smaller than that of the gallium arsenide 
tunnel diode. 

2. The Univibrator Circuit 
There are several possible variations of the basic txinnel 
diode univibrator shown in Plate II: Sometimes the bias volt- 
age is introduced on the other side of the inductor; occa- 
sionally there is a resistor between the tunnel diode and 
ground, and quite often all polarities are reversed. A 
typical output pulse is shown in Plate III with the letters a, 
b, c, d, and o corresponding to the same letters in Plate I. 
The voltage and current under quiescent conditions are given 
by the coordinates of point o in Plate I. The negative 



resistance region between b and d is unstable; consequently the 
tunnel diode switches rapidly to point c to accommodate the 
input current, thus producing the leading edge of the output 
pulse. Here precisely, the current may rise above c for large 
input pulses. The curve is then followed from c to d, thus 
producing the corresponding portion of the output pulse, where- 
upon the circuit relaxes to the operating point, o. The 
wiggles in the tail of the pulse are due to reflections within 
the circuit and oscillations about the operating point. 

The maximum voltage (or current) to which a pulse rises 
is the pulse height. If the initial slope of the pulse is 
small, and the top of the pulse is fairly flat, it may be 
difficult to measure exactly when the pulse starts and when 
it reaches its maximiim. Por these reasons the rise time, t , 
of a pulse is defined as the time for the pulse to rise from 
10% to 90% of the pulse height. If a trigger of minimal ampli- 
tude is" used with a constant current bias so\irce, for which the 
load line is horizontal, the rise time of a tunnel diode is 
approximately given by the relation t » ("V^tj" ^-d^^/^^-d" -^v^ ' 
where all quantities have been defined previously. The 
switching time, which is determined mainly by the junction 
capacitance sind somewhat by the amoxint of charge available 
from the trigger, of a GaAs tunnel diode with a peak 

current- of 10 ma and a junction capacitance of 10 pfd is 

25 

reported to be approximately one nanosecond, ^ and this was 

verified on a Tektronix Tunnel Diode Rise time Tester. With 



the tunnel diode connected into a iinivibrator circuit, the 
rise time of an output pulse is somewhat greater, typically 
aroxind two nanoseconds or less. 

In a manner similar to the above, the decay time, t^, is 
defined as the time for the pulse to decay from 90% to 10% of 
the pulse height. The pulse width is the full width of the 
pulse at half maximum, FWHM. Because the circuit is under- 
damped, the pulse overshoots equilibrium during decay, and 
the recovery time includes this overshoot. The minimum time 
between two input pulses which will cause the univibrator to 
be triggered twice when the second pulse is 10% above threshold 
is the pulse pair resolution time, t . In the simple tunnel 
diode univibrator the expression t = 14(FWHM) is valid over 
a fairly large range. ' Whetstone has shown that by replacing 
the 13 ohm load resistor with a high conductance diode and 
resistor combination, the recovery time of the circuit can be 
controlled so that pulse pairs well under ten nanoseconds 
apart can be coxinted, but the simpler circuit was considered ' 
adequate for our purpose. 

The output pulse height is a function of the inductance 
in the circuit. Results from four units, which are plotted in 

IlQ 

Plate IV, differ from those obtained by Whetstone for several 
reasons. The pulse height was measured not across the tunnel 
diode itself but at the point where the pulse entered the 
coincidence circuit; therefore the pulse heights were smaller 
than those of Whetstone. The inductance was somewhat difficult 



8 



to measure due to its small magnitude. The inductance plotted 
is that of the inductor, as measured with a Tektronix IJO L-C 
Meter, and does not include the stray inductance. The output 
pulse height also varies with the input pulse height, and is 
some constant plus about 5% of the input pulse height. This 
dependence is greatly reduced hy feeding the output from the 
first univibrator to a second univibrator to produce a stand- 
ardized pulse which is almost independent of the input pulse 
height. Over a range of several volts the variation in 
pulse heights from a two stage unit was just barely perceptible 
on a Tektronix sampling scope. There is another advantage in 
using a two stage unit. For a wide input pulse the first 
stage may produce a train of pulses, but with careful adjust- 
ment of the bias on the second stage, it will fire on only the 
first pulse in the train due to the fact that the first pulse 
of the train has the greatest pulse height. 

A complete circuit as shown in Plate 7T has two discrimi- 
nator units, each with two tunnel diodes, and a coincidence 
unit with two tunnel diodes. The inductor in the final uni- 
vibrator has been chosen larger than any of the other induct- 
ances so that the final output pulse is wide enough to be 
accepted by succeeding equipment. Through the complete circuit 
there is a propagation time which measures about eight nano- 
seconds but depends on the bias settings. The bias on any 
univibrator stage must be set low enough so that a pulse from 
the previous stage will trigger it but high enough so that 



feedback pulses from succeeding stages will have no effect. 

Of the several input circuits tested, it was difficult to 
ascertain which one was really best since all were able to 
fire the univibrator. Because of the threshold nature of the 
circuit, the univibrator either fires or does not fire, so 
simply looking at the output does not give a measure of the 
quality of the input circuit. A transistor input circuit was 
constructed which ultimately gave good results, but as there 
were some difficulties in getting the bias voltages for the . 
transistor initially adjusted, and as the amplification proper- 
ties of the transistor were not needed, this circuit was 
regarded as producing unnecessary complications. Pulse trans- 
formers constructed from ferrite cores according to informa- 
tion obtained from Whetstone were used in the input circuit 
and gave acceptable results, but perhaps due to the use of 
wide input pulses, slightly greater input pulse heights were 
required. A simple capacitor input circuit was also tested 
and gave good results with capacitances from 35 uuP to 100 
times this value. 

3. Detection of Coincident Pulses 
Consider two standard pulses, identical except for time 
of occurrence, which are added linearly and fed to a coin- 
cidence circuit. Because the coincidence circuit is set so 
as not to trigger on a single pulse height, it will not trigger 
\inless the pulses overlap and the sum of the overlapping 
pulses at some time is greater than the single pulse height. 



10 



This can occur only if the pulses are coincident to within a 
time of less than the FWHM of a single pulse. In practice the 
coincidence circuit must be set to trigger at a threshold 
greater than once hut less than twice a single pulse height. 
The same principle can also be used to determine if three or 
more pulses are coincident within a certain time interval. 

Other methods of time analysis than that of pulse overlap are 

q 
sometimes used.^ 

To measure the resolving time a series of 50 ohm cables 
of different lengths were made and the time for a pulse to 
propagate through each was measured with a Tektronix sampling 
scope. A single source fed the inputs of the coincidence 
circuit through different cable lengths. By selection of the 
cables it was determined that the resolving time of the cir- 
cuit in a typical configuration was between 2.5 and 3 nano- 
seconds, which was consistent with the standard pulse widths. 

4. Zero Crossover 
Complications arise from the statistical nature of photo- 
multiplier pulses, the problem being especially acute in the 
case of slow pulses from Nal(Tl) detectors.^ The question 
arises as to which part of the photomultiplier pulse gives the 
best time measiirement. When a quantum of radiation strikes 
the cathode, a photoelectron is ejected. The electron is 
accelerated to the first dynode where more electrons are 
ejected, and so on. Since the electrodes have finite dimen- 
sions, the paths of different electrons vary slightly in length. 



11 



and the final anode collects a swarm of electrons which, arrive 
within some interval of time. 

The trailing edge of the pulse is highly dependent on the 
circuit <. j not useful for timing pxirposes. The leading 

edge of the photomultiplier pulse tells only when the first 
few electrons arrive at the anode, but we seek the average time 
of arrival, which is approximately at the time of the maximum 
height of the pulse. As the top of the pulse may be fairly 
flat, the time at which it reaches a maximum may be difficult 
to ascertain. The negative photomultiplier pulse can be dif- 
ferentiated thus producing a pulse which is initially negative 
going and then positive going. If the differentiating circuit 
is properly designed, the zero crossover between the negative 
and positive going parts of the differentiated pulse will be 
a good measure of the time of the maximum of the photomulti- 
plier pulse. The zero crossover signal may also be obtained 
by the clipping stub method or the LC tank circuit method. 
Some circuits are so sensitive that they can detect a single 
photoelectron. In practice our coincidence circuit does not 
respond at exactly the zero crossover but very soon afterwards. 

5. Noise and Feedthrough 
With a single input signal of 100 mV the feedthrough at 
the output of the second stage is 0.8 mV, while a single 500 mV 
signal gave a feedthrough of 1 mV. Noise generated in the cir- 
cuit was certainly less than 1 mV. Since the output pulse for 
coincident singles had a height of several hundred millivolts, 



12 



noise and feedthrough were negligible and did not upset suc- 
ceeding circuits. 

An important consideration is the amoxmt of noise and 
feedthrough. at the coincidence diode. A 100 mV signal at the 
input produced a signal of 2 mV at the coincidence diode due 
to noise and feedthrough, and varied almost linearly with 
increasing input signal. As a smaller input signal produced 
smaller feedthrough, and as the standard pulse had a fixed 
height, it appeared best to use the least amount of amplifica- 
tion needed before applying the signal to the first tunnel 
diode. The effect of large signals was to increase the resolv- 
ing time of the circuit. The circuit operated most effec- 
tively when the input signal was not too much above threshold, 
say 30% at most. For very severe overdrive the coincidence 
diode might be expected to fire on a single pulse. With the 
circuit set to give an output pulse for coincident 100 mV 
pulses, a single input pulse of 20 volts would not fire the 
coincidence diode. 

During one period of counting it was found that many 
spurious pulses were registering on the scaler at odd intervals. 
While observing the scaler it was determined that these, pulses 
always occurred when an electric desk calculator was being 
used in the same room. Thus the circuit was not adequately 
shielded from line transients. 

6. Slewing 
The "slewing" or "walking" characteristic is defined as 



13 



the time variation of the output pulse as a function of excess 

46 
of the input pulse over threshold. The threshold must be 

set at a few millivolts above zero, otherwise the circuit will 
oscillate. For this reason the circuit can never really trig- 
ger at the zero crossover but will trigger a very short time 
afterward when the signal has risen to threshold. As the 
amplitude of the input signal is increased, the slope at the 
zero crossover increases; because the signal reaches threshold 
sooner, the tunnel diode would be expected to fire sooner. 
This is the case for a positive pulse, but for a zero cross- 
over pulse another factor comes into play. The negative por- 
tion of the pulse drives the operating point of the tunnel 
diode down the positive resistance part of the characteristic 
curve, and it would seemingly take longer for the current to 
rise to the peak point. Thus there are two competing effects. 
The greater slope of a higher amplitude input pulse tends to 
make the tunnel diode fire sooner, while the larger negative 
portion of the input pulse drives the tunnel diode down from 
the operating point and tends to make the tunnel diode fire 
later. The result is a U-shaped slewing char-acteristic as the 
tunnel diode fires late for both small and large zero crossover 
input pulses. Plate VIII shows the slewing characteristic as 
a fxxnction of per cent of threshold. The slewing characteristic 
of a commercial tunnel diode coincidence circuit is shown in 
Plates XI and XII. These results sire not quite as good as those 
which have been reported, even though we used a germanium 



14 



rather than a gallium arsenide .^el diode at the input stage 
to improve the slewing charact .stic. 

7. Effect Temperature 
At higher temperatures -.3 effect of intrinsic carrier 
concentration, the result of thermally created electron-hole 
pairs, becomes large in comparison to the effect of impurity 
atoms. In the tunnel diode this results in a changing of both 
the peak current and the peak voltage. In a germanium tunnel 
diode the decrease in peak voltage with increasing temperature 
is about 0.11 mV/C^. In our circuit the tunnel diodes were 
mounted in a position which allowed adequate ventilation to 
reduce temperature changes to a negligible amount. 

8. Jitter 
The jitter of the threshold on the input stage is approxi- 
mately 2 mV for the germaniiim tunnel diode. The minimxim usable 
threshold without sacrificing stability is approximately 60 mV, 
but usually the threshold voltage is set at 100 mV or higher. 
The jitter of the threshold voltage results in a time jitter 
of the firing of the tunnel diode. This time ji't'ter was 
determined to be a small fraction of a nanosecond, and ^or 
the present equipment is small enough to neglect. 

9. Nanosecond Systems, Inc., Equipment 
In the fall of 1964 equipment was p\irchased from Nano- 
second Systems, Inc. The circuit employed is that of Whetstone^^ 



15 



with improvements. ' It was found to behave quite similarly 
to the previously constructed equipment. It possessed many of 
the same difficulties such as double pulsing on wide input 
pulses. One good feature was that the commercial equipment 
was much better shielded and did not respond to line transients 
from the previously troublesome electric desk calculator. 

The resolving time of the ITajiosecond circuit was measured 
in two different ways for various settings of the knob labeled 
"logic" on the coincidence unit. Table I shows the data 
obtained by feeding signals from a pulser to two discriminators 
through different delay lines. The range in time over which 
one signal could be varied relative to the other was then 
taken as twice the resolving time of the circuit. As a check 
the linear sum of the two standard 410 mV pulses entering the 
coincidence circuit was also viewed on the sampling oscillo- 
scope. Table II shows the maximum height of this sum. with 

Table I. Resolving time by the direct delay method. 



Logic knob Resolving time in 

setting nanoseconds 



3.60 16.3 

4.00 14.5 

5.50 5.5 

7.00 2.0 

7.50 .75 



various delays in the circuit. The logic knob on the coinci- 
dence unit is actually a threshold setting knob although it is 



16 



not directly marked in millivolts. Suppose a particular 
setting of the logic knob, say 5»00, corresponds to a threshold 
of 610 mV. Prom Table II we see that one pulse could be varied 
in relative time from 28 to 48 nsec, which would give a resolv- 
ing time of 10 nsec* The resolving time at various logic 
setting was also determined by the chance coincidence method 
from the data in Table III. The right hand column was calcu- 
lated from the formula t = N^/(2 N^K2)» where N^ and N^ are 
the single counting rates and IT is the chance coincidence 
counting rate. As shown by the plot in Plate IX the results 
of the two methods are in excellent agreement. 

Table II. Maximiim height of two overlapping standard pulses. 



Delay between pulses Maximum height in 
in nanoseconds millivolts 

10 410 

16 420 

20 510 

28 610 

30 660 

36 810 

40 790 

48 610 

50 550 

56 420 

60 410 



17 



Table III. Resolving time by the chance coincidence method. 



^ . „ . r, ^- Count rate Coxint rate Resolving 

Logic Coin. Counting ^^^^ ^^go from pulser time 

setting counts time ^^^^^^ (^^^^^ {nzec) 

3.60 1684 30.00 min 204 21? 

4.00 660 30.00 min 204 21? 

5.50 312 30.00 min 205 277 

7.00 1423 12.00 hr 203 783 

7.50 801 17.75 hr 205 ^58 



251 682 


16. 5 


251 592 


12.9 


251 587 


6.1 


251 205 


1.15 


251 789 


0.435 



III. THEORY OF DIRECTIONAL CORRELATION 
1. The Directional Correlation Function 

Directional correlation measurements involve the coin- 
cidence counting rate as a function of the angle between the 
propagation vectors of two cascade radiations. Many treatments 
of the subject are found in the literature, but we will give 
only a brief discussion of the theory since this paper is 
mainly concerned with the experimental problems. 

For convenience we take the direction of propagation of 
the first gamma ray as the polar axis of spherical polar 
coordinates. The angular distribution can be most appropri- 
ately described by a series of spherical harmonics: 

WC0,0) -^aj^y^(O,0), (1) 

L,m 

where WC©,0) is proportional to the coincidence counting rate 
as a function of the direction of propagation of the second 
gamma ray. 

For randomly oriented nuclei and the special polar axis 



18 



we observe that the distribution must be independent of the 
azimuth angle. In this case m=0, and the series may be 

simplified: 



W(0,0) -Ea^-Q I°(9,0) - W(©) 



W(0) = EAl P^Ccos G) (2) 

L 

where G is the angle between the propagation vectors. A gen- 
eral derivation with an arbitrary orientation of the coordin- 
ate system gives the same result. 

Two more simplifications can be made. If the transitions 
involve states of definite parity, only the even Legendre 
poljmomials appear. Since the transitions are between 
eigenstates of angular momentum, the maximum value of L is 
limited by certain triangular inequalities: 

L/2 = Min(L^, I, Lg) (5) 

where L, and Lp aire the angular momenta of the radiations and 
I is the angular momentum of the intermediate state. 

Of course this series could be expressed in terms of 
another set of functions such as powers of cos G, but a par- 
ticular advantage of Legendre polynomials is that each coeffi- 
cient can be split into two factors, each of which depends on 
only one of the transitions. For pure transitions, 

^L " ^L^^l ^1 ^i ^^ ^L<^^2 ^2 ^f ^> ^""^ 

where I. and I-. are the angular momenta of the initial and 
final states. These factors are certain combinations of 

r 



19 



Clebsch-GordorL and Racaki coefficients and have been tabulated 
in various references. Note that V(0) is symmetric with 
respect to an interchange of the order of the transitions. 
If a transition is mixed with a mixing ratio, d, defined by 

,2 intensity of (L + 1) ^5^ 

^ ' intensity of (L) ' ^^^ 

then the factors become 

——2 

Ve now derive these functions for gamma-gamma angular 
correlation in a more explicit but less useful form. Ini- 
tially a nucleus may be in any of the- 21^^+ 1 equally populated 
initial substates with spin I^ and projection quantum number 
M- . The first gamma ray is characterized by quantxim numbers 
L, and M, . For the special choice of polar axis M^ can be 
only +1 or -1. The intermediate state is characterized by I 
and M. These three sets of quantum numbers must satisfy the 
vector addition 

Xi^ . IL^ -*• n-MT (7) 

f 

From the definition of Clebsch-Gordon coefficients 
Thus the relative transition probabilities are given by 



u 



20 



and the relative populations of the intermediate substates are 

Ij (IL^^m^ ^i^i)^ (10) 

"l 
where we have summed incoherently since the phases are random, 

and the prime indicates that M, is only +1 or -1. The angular 

distribution of the first gamma ray is then 

A__j (IL^m, l.n.)\'^iQ) (11) 

where P^ (9) is some angular function characteristic of the 

■^1 

radiation. Por a radiating multipole of order L and projec- 
tion M 



p'^CO) = &(L-M)-H(M-1)3 |y/-^|2. 2H^1y^|^.[l(L^1)-H(M.1)]|y^^^|^ 
L 2 L CL + i; 

(12) 
Since the first gamma ray is observed only at 0=0, the corre- 
lation function is 



, , (IL^MM^ I^M^)^P^^(0)(I^L2M^M2 IM)^F^2(0) 

I^,M,M^ 1 -2 



M, 



M^= +1 or -1 

V ^^f -M , (13) 

2. The Ni^^ Cascade 
The standard nucleus for directional correlation measure- 
ment is Ni . The nucleus Co decay to Ni by negatron 
emission with end point energies of 0.513 Mev and 1.45 Mev with 
very nearly 100?^ of the transitions having the former end 



21 



60 
point energy. These transitions lead to a Ni nucleus in a 

2.5 Hev excited state as shown in Plate XIII. The groiind 

state is reached "by emission of two cascade gamma rays of 

energies 1.1? Mev and 1.35 Mev. The 1.55 Mev gamma ray has 

been established as the second transition by means of nuclear 

flourescence resonance experiments. The spin sequence is well 

established as 4(E2)2(E2)0 with both transitions being pure 

electric quadrupole. All three levels have positive parity. 

60 
The theoretical directional correlation function for the Ni 

cascade assuming the above spin sequence is 

W(0) = 1+0.102 PgCcos 0) + 0.0091 P^Ccos 9). (14) 

5. Analysis of Chance Rates 
Before comparing experimental data with the theory pre- 
viously described, several corrections must be made. We must 
first subtract chance coincidences between unrelated pulses. 
The analysis of chance coincidences is treated extensively in 
the literature.27.5'^.59.« 

In a double coincidence circuit the chance coincidence 
rate is easily shown to be 

N^ = 2 t(N^- lft^^^2"^t^ ^"^^^ 

where t is the resolving time, N^ is the true coincidence rate, 
and N^ and Np are the singles counting rates in the two chan- 
nels. In practice the true coincidence rate is always so very 
much smaller than the singles counting rates that the N^ terms 
may be neglected. 

In a triple coincidence circuit of the fast-slow type 



22 



there are several possibilities for chance counts. The various 
singles and coincidence rates are shoi«i in Figure 1. It is 
assumed that the singles rates, Kj and ITj^, from the fast 
discriminators are larger than the corresponding singles rates, 
N, and Np, from the single channel analyzers. 



PM 1 



PM 2 



disc 1 -1 



disc 2 



^1 



N 



II 



Ni 



SCAl 



JJ« 



SCA 2 



fast 
coinc 



slow 



coinc 



t3 



N 



Fig. 1. Block diagram of a fast-slow coincidence circuit. 



For simplicity it is assumed that the coincidence circuits 
operate in the following manner. A single pulse input to the 
fast circuit produces a rectangular pulse of width t^. If two 
pulses from the two fast inputs overlap at any time, an output 
pulse is produced. The slow circuit works in a similar manner 
except that pulses from all three inputs must overlap at some 
time in order to produce an output pulse. 

The chance coincidence rate "between N, and 1^2 o^l7 is 
given by 



-1 



25 



1T(1 S: 2) = (t^+ t2)M2 (16) 

with, an average overlap time of t,t2/(t^+ tg)* The triple 
chance coincidence rate is then 

= (t^t2+ t^tj+ t2t^)lT^ir2N5 (17) 

Por t,= tp" t^=» t, the preceding expression is 

N^(lSi2Si5) - 3 t^ ^^^211, (18) 

The fast coincidence rate, N"^, can be split into several parts; 

where N. is the true coincidence rate between gamma rays of 
the proper energies, N^., is the fast coincidence rate where 
only the first gamma rays have the proper energy, N^2 1^ "''^^ 
rate with only the second gamma rays having the proper energy, 
N» is the fast coincidence rate between gamma rays, neither of 
which has the proper energy, and the la^t term in (19) is the 
chance coincidence rate. 

The triple slow coincidence rate, N, can be divided into 
eight parts: 

N - 11^12^ Nfi(2)-^ ^f(l)2^ ^f(l)(2)* ^(f)12^ ^(f)l(2) 

*^(f)(l)2-' ^(f)(l)(2) (20) 

where parentheses indicate a component which is in slow coin- 
cidence but not fast coincidence with the other components. 
As an example l^fn /•p') ^^ *^® coincidence rate between N^, and 
a gamma ray of energy E2, the latter being in slow but not 



r- 





24 


fast coincidence with the former. The nature of the various 


terms is easily deduced from the following formulas: 




"^1(2) - 2 t N^^Nj 


(21) 


"f (1)2 - 2 * ^tzH 


(22) 


''f(l)(2) - 5 *^ V2''f 


(23) . 


''Cf)l(2) • ^2 Vii%) 2 t Hg 


(24) 


K(f)(l)2 - ^2 tAH2) 2 t S^ 


(25) 


K(i:)(l)(2) - 5 *^ V2 2VAI 


(26) 


Equation (20) now becomes 




N - 1T^+ 2 tCNf;L^2"' ^f2^1^ + 5 t^ N^IT2^^+ 2 t^3_lT2 




+ 4 t^t N]_N2^^I+ %I^ + 6 t^ *f^lN2^I^II- 


(27) 


4. Experimental Considerations 




The singles counting rate in either detector is 




N. . Mp.e.n. 


(28) 


where M is the source strength in disintegrations per unit 


time, p. is the probability that a disintegration will 


lead to 


the gamma ray of interest, e^^ is the efficiency of the 


detector, 


and n. is the solid angle subtended by the detector in units 


of 4 ff. The accidental or chance counting rate in the 


coin- 


cidence circuit is 




No" 2 t e^NilT2 




- 2 t e^Mp^e^n-LMp2e2n2 


(29) 


where e is the efficiency of the coincidence circuit, 
c 


The 



25 



true coincidence rate is 

N^= M PiP2eie2e^nin2 K(9) (50) 

where K(0) is proportional to the directional correlation 
function and has an average value of unity. The true to 
chance ratio at any angle is 

^ ^ M PlP2^c^l^2^1^2^^Q^ (51) 

^c ^ ^ ®c^ Pi^i^l^ P2e2^2 
so that the average value is 

CWave - ^/^2 t^). (32) 

The actual counting rate is the sum of the true and chance 
rates from which the chance rate must he subtracted. In order 
for the chance rate to contribute only a small error to the 
results the true to chance rate should be, as high as possible. 
In practice there is a lower limit for the resolving time of 
the coincidence circuit due to electronics. Furthermore if H 
is small, the counting rate will also be small, but it is 
desirable to have the singles rates many orders of magnitude 
above the background rate. A good compromise is to have the 
true to chance rate between five and twenty. 

As a numerical example, let N^/N^» 5 and t^ - 25 nsec. 
Then from (52) 

M - lT^/(2 t^N^) « 0.1 mc (35) 

Also, let e^^- e^- 0.05, n^=^- 0-01, and e^- 1. From (29) and 

(50) 

N a 4.5/min 
c 

N^= 21.5/min (5^) 



26 



Suppose we wish to know the counting rate to within the frac- 
tional statistical error E, and there is no error in the 



tine T. 
e2 



»t*c* %^/''K ^> 



For E » 1%, T » 600 min at each angle. Por E » 9.1%, then 
T = 60 000 min at esich angle. As will be shown in the next 
section, for coimters which subtend a finite solid angle, the 
directional correlation function is attenuated. For this 

example 



U(9) = 1 + 0.100 P^Ccos ©) + 0.0086 P. (cos 9) 



(36) 



with a few values being 
WC90^) = 0.9532 
W(135^) - 1.0215 

W(180^) » 1.1086 (37) 

In order to propagate the errors we need White's matrix: 
42 56 7^ I^O0°)\ /Aq\ 

I05 I -90 40 50 
48 -96 48 



7 



/ N(155°) 
1 N(180°) 



/ 



\^4/ 



(38) 



Suppose we wish to determine A^ to within ten per cent,, then 
from (58) the counting rate of true coincidences at each angle 
must be about 0.15% correct. From (35) the counting time at 
each angle is twenty days I If both counters accept both gamma 
rays, only half this time is required. We see that a tremend- 
ous amount of time is required to collect directional 



27 



correlation data. 

5- Correction for Finite-Size Detectors 
When the directional correlation is observed experi- 
mentally, it appears "smeared out" according to 

W(0) = — (39) 

/y dn^dn2e^e2 

The efficiency, e, of a detector has been treated in several 

17 
ways. Feingold and Frankel ' use an expansion in spherical 

hairmonics for the efficiency of an arbitrary detector 

e(0,0) = C^^^''^Lml^(G,0) (^) 

L,m 

For a circularly symmetric detector as shovm in Figure 2, this 

expression simplifies since e is independent of 0. 

eCG) = Ea^ P^-Ccos Q) (41) 

The expression which Rose^' uses for the efficiency is e » 1 - 
exp(-kx), where k is the absorption coefficient for the gamma 
ray energy of interest and x is the detector thickness at the 
angle of incidence of the gamma ray. For a circularly sym- 
metric detector 

X » t sec b for 0^ b ^tan"-'- [r/(h+t)] - b' 

X = r CSC b - h sec b for b' ^ b ^ tan" (r/h) (42) 
Since the directional correlation function is most often 
expressed as a series of Legendre polynomials, we must perform 



28 



detector 1 




detector 2 



Figure 2, Angles involved in correction for finite- 
size detectors* 





• 


29 


integrations of the form 






I ^ J^y dS\^dS\^e^e^jJ^QQs ©')• 




(43) 


Ve express P^Ccos 0') in terms of b^ and 0" "by 


means of 


the 


addition theorem for spherical harmonics. 






+L 

P^Ccos e-) = 5^C YS'(t,,0) YS(e",0") 

m=-L 




(44) 


Let X, "be independent of 0^ so that only terms 


with m»0 


con- 


tribute to the integral. Then 






P^Ccos 0') - ^ YO*(bi,0)Y2(Q",0") + ... 


» 




P^(cos 0') - Pj^Ccos b^) P^Ccos Q") + ..., 




(45) 


and the integral I becomes 






/ d0^y^sin bj_db^Pj^(cos 'b^)e^/j'^0^3±a, Q' 


"d0"Pj^(cos 0")e2 






(46) 


The efficiency of the second detector is most < 


easily expressed 


in terms of angles measured from the Zp axis. 


Ve apply 


a 


rotation matrix and drop the m^O terms which do not contribute. 


This gives 




i 


P^Ccos ©") - Dq5(0,©,0) P^Ccos b2) + ... 






» Pj^Ccos 0) Pj^Ccos bg) + ... 


i 




The integral I becomes 






I - 4ii^ Zjil) Jl(2) Pj^Ccos 0), 


' 















30 


where 








- 




Ji,(i) 


1 « / sin b,db^P-j^(cos l^i)©]^ 










Jl(2) 


( = / sin b2db2PT (cos b2)e2« 






(47) 


The 


experimentally observed directional 


correlation 


function 


then 


becomes 






• 




W(0) 


= 1 + (^22^2^2^^°^ ®^ "^ Q44^4?4 


(cos 0) + ., 


• • > 




where 




f 








Q22 = 


. Q2CI) ^2^2^ 










Q2CI) 


1 = J2(l) / JqCD. 






(48) 


As an approximation we assume that e is 


constant up 


to 


b^ and 



zero 


thereafter. 










h- 


♦ 

^b 
/ Pt-(cos b)e sin b db 










Jo- 


ey sin b db « e(l - cos b ) 










^2- 


e /P2(cos b) sin b db » e cos 


b^ (1 - cos2b^)/2 




J4" 


e /p^(cos b) sin b db 










- 


2 2 
e cos b (1 - cos b )(7 cos b 


- 3)/8 




(49) 


Then 


the attenuation factors for similar 


detectors < 


ire 






Q22 = 


' (Q2)^ 






1 




^2- 


COS b^ (1 + cos b )/2 










«W " 


■ (Q^)^ 










«4" 




cos b^ (1 + cos b )(7 cos b - 


5)/8 




(50) 




\- 


tan~-^(r/h). 









These are the formulas given "by Frauenfelder and Steffen 

52 

As pointed out by Lawson and Frauenfelder, the 



31 
18 



attenuated directional correlation fxinction is of the form 
derived above only for centered point sources and circularly 
symmetric detectors. Otherwise a mixing of the different At 
terms occurs. 

IV. MEASDEEMENT OF THE Ni^^ DIRECTIONAL CORRELATIOIT 

1. Apparatus 
A source, roughly of strength 0.1 mc as will be shown 
later, was prepared by evaporating a Co Clp solution inside a 
carbon cylinder of outside diameter 5/16 inch and inside 
diameter 3/52 inch. The wall of the carbon cylinder consti- 
tuted an absorber of 0.575 gm/cm , which was more than suffi- 
cient to absorb all the beta particles from the 0.515 MeV beta 
decay, for which only 0.086 gm/cm was needed. The total 
absorbing material consisting of the cylinder wall and the 
cover on the Nal crystal was not quite enough to absorb 

completely all beta particles from the 1.^ MeV beta decay, 

2 

which would have required 0.7 gm/cm , but no additional 

absorber was introduced since this decay is only 0.01% as 
intense as the 0.515 MeV decay. 

The fixed detector was a ITal(Tl) scintillation crystal 
2 inches by 5 cm thick placed 9»2 cm from the source, and the 
movable detector was a Nal(Tl) crystal 1 5/^ inches by 5 cja 
thick placed 8.4 cm from the source. Availability was the 



32 



only reason for choosing different sized crystals. Each 
crystal was mounted on an EGA 65^2 photomultiplier tube 
operated at 1100 volts and placed on a specially constructed 
circular table with the source at the center. Each crystal 
was surrounded by a very thick conical lead shield. 

Fast coincidences between pulses from the photomulti- 
pliers were detected in the Nanosecond Systems equipment and 
fed to a triple slow coincidence iinit built by David 
Draegert. Photomultiplier pulses were also directed through 
a Nuclear Data 500 dual single channel analyzer to the slow 
coincidence unit. The fast-slow coincidences were counted on 
a scaler. Simultaneously singles counts passing through the 
Nuclear Data 500 analyzers were counted on scalers. 

2. Determination of Resolving Times 
A 1.5 uH inductor was placed in the fast coincidence logic 
circuit. The resolving time of the fast circuit was deter- 
mined by the chance coincidence method. Two rather long inins 
were taken giving the following: 
Nj » 64 927 808 / 856 min 

N-j-j = 18 187 418 / 856 min 

N » 1205 / 856 min. 
o 

Using (15) the resolving time of the fast circuit is 

*f " \/^^ Vll^ " ^^'^ i ''^ ^®®*^* 

The logic setting used was 5 •70. Note that the graph in 
Plate IX cannot be used here because the inductors were not of 



55 



the same value, but from Plate H and the graph of Plate V, 
we can conclude that the result is entirely consistent with 
previous measurements. This resolving time will be used to 
calculate expected chance rates. 

The resolving time of the slow coincidence unit was 
determined by the same method. 

Nj_ = 505 5^ / 5 mill 

N^ = 9^ 612 / 5 min 

N = 507 / 5 min 
c 

The resolving time of the slow unit was then 

t = 1.60 + .09 usee. 
This was consistent with oscilloscope observations of the 
waveforms in the circuit which showed the, pulses fed to the 
slow coincidence unit to be about 1.5 usee wide. 

5, Determination of Chance Rates 
The measured true plus chance slow coincidence rate 
between K, and Ng was about 105 cpm of which 91 cpm was the 
expected chance coincidence rate and 74+8 was the measured 
chance rate. The measured true plus chance fast coincidence 
rate between Nj and ^-^j was about 55 cpm of which 5*7 cpm was 
the expected chance rate and 5.8 ± 'S cpm was the measured 
chance rate. Prom this a very rough calculation can be made 
for the source strength from (52), which gives M - 0.07 mc. 
At this stage, however, the data are much too rough to give 
more than an order of magnitude. A more precise measurement 



54 



showed the source strength to be 0.13 mc. 

Measurements were taken of the various counting rates 
and approximate calculations were made in order to obtain some 
idea of the relative contributionn to the chance counting rate, 
The data used were: 

N^ - 20 000 cpm N, » 8.? cpm 

^2 " 100 000 cpm isr^^ - 1.9 cpm 

% = 35 000 cpm lf^2 -10.7 cpm 
Njj = 120 000 cpm 

t = 1 usee t^ - 25 nsec. 

Calculation of the terms in (27) gives: 

2 t(lT^^lT2+ ^£2^^) - 0.007 cpm ^ 

3 t^lT^N2N^ = 0.00000016 cpm 

4 t^t N3_N2(N-|.+ ITj-j.) - 0.009 cpm 

6 t^t^N^N2lTjNjj = 0.00000016 cpm 

2 t^N^N^ =1.7 cpm. 

It was apparent that the last term made the only significant 
contribution, so all other terms were neglected. 

4. Determination of Correction for 

Finite-Size Detectors 

22 

A Na source was mounted on the coincidence table and 

coincidence rates were taken at nine points about 180°. The 
annihilation radiation consists of coincident 0.5 MeV gamma 



35 



rays at an angle of 180°. The chance coincidence rates were 
subtracted, and the data were plotted. A smooth curve was 
drawn through the points. The half width at half maximum was 
measured to be b « 5° 15' + 5»5'' Each detector was then 
assumed to subtend a cone of solid angle 0.027 steradian. 

Note that the energy of the annihilation radiation was 

60 
not the same as that of either gamma ray from Ni . A better 

but more complicated procedure for finding the half-angle b 

experimentally would have been to use a well collimated beam 

of the same energy as each of the gamma rays. 

Solid angle correction factors were calculated from (50) . 

Q2Q2= 0.9823 + .0005. 

(^Q^= 0.9^20 + .0001. 

60 
The theoretical directional correlation function for the Ni 

cascade (14) is attenuated to 

W(0) - 1 + 0.100 P2(cos 9) + 0.0086 P^(cos 0) 

5. Reduction of Error at 135° 
Since there are no odd Legendre polynomial terms in the 
directional correlation function, it is symmetric about 90° 
and 180 . This means that the slope is zero at these two 
angles thus making them ideal experimentally since a small ' 
systematic error in the angles contributes very little error 
to the measurement. At 135° > the other standard angle, the 
slope is not necessarily zero. Indeed in the case of Ni 
this is quite neair the point of inflection where the slope 



36 



is maximum, so that a small error in the angle setting at 135° 
may contribute greatly to the error ir, the measurement. 

It was thought that a systematic error at 135° could be 
eliminated by taking part of the me-' ^urements at 135° and part 
at 225 and adding the counts as if they were all taken at 
135 • It was found that the counting rates at these two 
angles differed by less than the statistical variation after 
a total of almost 20 000 counts had been taken; nevertheless 
these measurements were still considered to provide an addi- 
tional check on the operation of the experimental apparatus. 

There are, of course, two other angles that might be used 
in this method, namely 45° and 315*^ » but these cannot be 
realized experimentally at the present time because of the 
apparatus . 

60 
6. Analysis of ITi Correlation Data 

60 
The Co source was centered on the coincidence table so 

that the counting rate in the movable detector varied not more 
than one per cent over the range of angles used. A singles 
spectrum was observed from each detector. The discriminators 
on the fast unit were set just above the Compton edge of the 
1.17 MeV peak. The single channel analyzers of the slow unit 
were similarly set. In this way both photopeaks in both 
channels were used, and the true coincidence rate was effec- 
tively doubled. 

After the initial setup procedure a series of runs with 
equal to 90°, 135°, 180°, 225°, and 270° were taken in which 















57 


Table IV. 


Directional 


correlation 


data. 






Angle 


Time 


Co\)nts from 


Counts 


5 from 


True plus 


between 


in 


fixed 


movable 


chance 


detectors 


min 


detector 


detector 


counts 


90° 


15.88 


269 


524 


900 


005 


84 


135^ 


15.69 


264 609 


900 


000 


71 


180^ 


15.97 


270 


537 


900 


001 


82 


225° 


14.25 


288 


582 


900 


000 


74 


270° 


14.08 


288 


Oil 


900 


001 


79 


225° 


14.28 


288 


597 


900 


001 


72 


180° 


15.92 


281 


750 


900 


000 


65 


180° 


14.05 


284 


747 


900 


000 


77 


155° 


15.7^ 


276 491 


900 


001 


74 


90° 


15.94 


278 


090 


900 


004 


62 


90° 


15.90 


276 


871 


900 


002 


73 


155° 


15.74 


271 


586 


900 


000 


81 


180° 


15.90 


275 


289 


900 


001 


87 


225° 


14.18 


281 


495 


900 


000 


75 


270° 


15.82 


289 


448 


900 


001 


65 


225° 


14.21 


295 


566 


900 


001 


65 


180° 


15.92 


289 


850 


900 


002 


74 


155° 


15.87 


288 


255 


900 


002 


68 


90° 


14.24 


288 


115 


' 900 


002 


62 


155° 


14.25 


288 


446 


900 


000 


84 


180° 


14.72 


298 


850 


900 


001 


90 


225° 


15. 06 


506 


215 


900 


001 


67 


270° 


14.71 


512 


021 


900 


002 


75 


225° 


14.92 


314 


948 


900 


001 


79 


180° 


14.56 


308 


442 


900 


001 


80 


155° 


14.28 


300 405 


900 


001 


92 


90° 


14.55 


297 


049 


900 


001 


83 


155° 


14.17 


290 


466 


900 


002 


95 


180° 


14.54 


297 


857 


900 


001 


86 


225° 


14.86 


305 


828 


900 


002 


89 


180° 


14.44 


296 


997 


900 


002 


97 


155° 


14.17 


290 


520 


900 


005 


97 


90° 


14.14 


290 


789 


900 


001 


84 


135° 


14.11 


289 


007 


900 


005 


75 


180° 


14.42 


296 


548 


900 


001 


77 


225° 


14.73 


503 


552 


900 


002 


85 


• 



58 



a total of 67 752 coincidence counts were obtained in 8275.23 
minutes of actual counting time. Intermittently between these 
runs various checks were made. Singles rates were counted for 
short intervals, waveforms ■• .-e checked on an oscilloscope, 
and spectra were taken on a multichannel analyzer. 

Table IV shows a portion of the data. In order to reduce 
the effect of a slightly non-centered source, it is standard 
practice to normalize the coincidence rates by dividing by the 
appropriate singles counting rates in the movable counter for 
each angle. It was thought that if data were taken up to a 
preset number of counts in the movable detector, this normali- 
zation would be unnecessary for a quick check on the data. 
Note that the scaler had some difficulty in turning off at 
exactly 900 000 counts. If the counting rate in the movable 
detector is calculated for a particular angle, it is seen that 
in general the counting rate decreased with time. These data 
were all taken within a ten hour period, which. is extremely 
short compared to 5-3 years, the half-life of Co^°. The change 
in counting rate was apparently an electronics instability. 

60 
Table V. Ui directional correlation data. 



angle time 0^ counts 0° counts *^® + 

chance 



,?^o-270o 15^S.40 min 30 676 393 134 8^3 117 11 684 
}ilo-^^^ S§§^-J^ ^^^ 50 870 273 217 082 741 19 835 
180 2725.66 min 53 951 282 239 154 205 23 231 



39 



The wide variation in the number of counts in the fixed 
detector is expected since the statistical error is composed 
of two parts. The fixed detector counting rate varies statis- 
tically in itself, and the time varies statistically due to 
taking a preset number of counts in the movable detector which 
has a statistically varying counting rate. 

The analysis went as follows: From the singles rates, 
the times, and the previous determination of the resolving 
time, the number of chance covmts was calculated for each 
angle and subtracted from the right hand column of Table V 
giving the numbers of true coincidence counts for each angle. 

The background rate in the movable detector, taken before 
the source was placed on the apparatus, was 665 cpm. The 
total number of expected background counts was calculated and 
subtracted from the number of counts from the movable detector 
for each angle. The resulting numbers were divided into the 
respective numbers of true counts to normalize for a slightly 
off center source. 

Errors were propagated according to the standard methods. 
Three numbers were obtained which were proportional to the 
normalized true coincidence counting rates: 



NOO'') = (709 ± 9) X 10"^ « N^ 

2 



1^(135°) = (756 + 7) X lO"*^ - N 



17(180°) - (817 + 7) X 10""^ - N, 

These numbers are proportional to the directional correlation 
function. A best constant of proportionality was determined 



40 



from a least squares fit to W(©) at the three angles. 

- -||- =C2(V k W(©^) )(-W(©i) ) 
Then we obtain k » 0.0758. The experimentally observed values • 
of the directional correlation function at the three angles 
are: 

W'(90°) » 0.9634 + .0117 

W'(155°) ' 1.0204+ .0099 

W'(180°) = 1.0988 + .0094 

These numbers were fitted by using White's matrix (38) 

to yield the experimentally observed directional correlation 

60 • 

function for the Hi cascade as 

W'(0) = 1.00 + (0.094 0.011)P2(cos 9) 

+ (0.0087 0.010)P^(cos 0). 
This is to be compared with the expected correlation function 
derived in section IV. 4, 

W(0) = 1 + 0.100 P2(cos ©) + 0.0086 P^(cos 0). 

Although the errors are rather large, the directional 
correlation fxinction can provide very useful information. 
Por instance, it can be shown that if the first gamma ray is 
mixed, the error in Ap limits the intensity of any M3 component 
to less than 0.03%. 



41 



V. CONCLUSION 
A tunnel diode fast coincidence circuit was constructed 
and the various properties determined to be as reported in the 
literature. A commercial circuit of a similar design was used 
in a directional correlation experiment, and the results were 
shown to agree with theory. It is concluded that with proper 
attention to the inherent problems, such as double pulsing, 
the equipment can be used in this type of experiment. 



42 



VI . ACKNOVrLEDGMENTS 



The author takes this opportunity to acknowledge and 
thank Dr. Louis Ellsworth for his guidance and assistance 
in completing this work. The author also wishes to thank 
Dr. Charles E. Mandeville and Dr. V. R. Potnis. 



44 



CvJ 



:n 



> 

> 


^ > 
> 


> 


a 
> 


+ 


CM ' 


1 

Q. 


iD 


D 


a 




N 


> 


> 




• 

O 




EXPLAlTAa?ION OF PLATE II 

The basic tunnel diode iini vibrator. 



-10 V 



46 



PLATE II 



500 ohm 



vVW 

13 ohm 




430 ohm 

-aAA/V 



GaAs r 

tunnel diode 
GE 1 N 3118 




germanium diode 
ID3-050 



EXPLANATION OF PLATE III 
Output pulse of a univibrator. 









if8 








, 








i 








; 








■3 








i 






O 




= 




( 


• 


—J 

0. 




( 






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time 


^<C 






, 

1 
{ 


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S;OA 







EXPLANATIOIT OF PLATE IV 

A plot of pulse height vs. 
inductance for the 'univihrator. 



EXPLANATION OF PLATE V j 

A plot of the base width of the I 

output pulse of a univibrator 
versus the inductance of the 
inductor shown in the circuit. 













52 












,■ 












> 


\ 


V O 


\ 


\ 


9 


- 


.60 .80 1.00 1.20 1.40 

inductance L (jjH) 


> 








\ 


/ 


.40 


u 

< 








\ 


. 


.20 




1 


1 


1 


1 1 \ 1 








■^ 


Cv) 


9 


00 to T 


CJ 








(D9S U) 


g 


mpiAA aseq 


■ 






' 






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J 



EXPLANATION OP PLATE VI 

The complete coincidence unit 
using six tunnel diode 
uni vibrators. 



E3a?LANAa}I0N OF PLATE VII 

(upper) Nanosecond Systems fast 
coincidence modules, 
(lower) Fast coincidence unit 
constructed in the laboratory. 



56 





EXPLANAOJIOIT OP PLATE VIII 

A plot showing the slewing 
characteristic of the coincidence 
\init constructed in the laboratory. 











58 

■■ 

3 
. 5 


> 

UJ 

< 

-J 
a. 


\ 


\p 


unit B 

threshold ^SCX) mv 

1 r . 


190 220 

• 




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. 


\ 


100 130 160 

per cent threshold 




1 


1 


1 t 


> 




o 


o 

CO 

(DOS U) 


o o c 













EXPLANATION OF PLATE IX 

A plot of the resolving time of 

the Nanosecond coincidence unit 

versus the setting of the logic 

knob. The inductor used was 1.00 uH. \ 

The resolving time 'was determined \ 

by two methods and it is seen that I 

the results of the two are in good 

agreement. 







60 


X 






u 

< 
-J 
CL 


direct delay 1 
chance nnethod I 


6.50 7.50 

• 




< / 


5.50 

setting 




/ 


logic 




/ 


4.50 




</5 


O 




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ID 
CO 




1 1 1 1 1 1 1 1 1 1 


• 


OcQ^"^C\JOoOCO\rCM 




(D9S U) 9UU|:^ 6U!A|OSGJ 





EXPLANATION OF PLATE X 

A plot showing the output pulse 
height of the pulse'r when fed 
into a 50 ohm line. 



EXPLAITATION OF PLATE XI 

A plot showing the slewing 
characteristic of the Nanosecond 
equipment including amplifier for 
various thresholds. The abscissa 
is the input pulse height before 
amplification. The curves are 
labeled with the setting of the 
discriminator knob* 



6k 



X 

u 

< 

_1 

a. 



u 

(J) 

(D 





I 




3, 


CM 


00 




CO 


L. 




o 


o 


o 




C 


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h 


3. 


£_ 


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in 


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o 

Cvi 



O 
10 



> 

E 



o en 




Q. 



O 






o 


o 


o o o o 


f^ 


to 


10 -^ CO CM 

(Dss u) v;iL|s auui; 



EXPLANATION OF PLATE XII 

The same data used in Plate XI 
are replotted with 'the ahscissa 
as per cent of threshold. 



66 



X 



U 







I 




CM 


D 




£_ 


00 




o 


vO 


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< LU 



Q. Ul 



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6 




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/ > 


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in / 


o 


K/ 


o 



I 

o 
to 



L 

O 
10 



1 

o 



L 

O 
CO 



-JL 
O 
CM 



O 



o 
o 

CO 



o 

in 



o 

o 

CM 



O 



o 

- o 



EXPLANATION OP PL/iTE XIII 

60 
The decay scheme of Co showing 

the two gamma rays of interest. All 
energies are in Mev. Although the 
spins of some states of Ni^O are still 
iincertain, the energies, spins, and 
parities of the states involved in 
the directional correlation measure- 
ment are well established. 



PLATE Xlll 



68 



,(+) 



v3. /2. 



S'^ w Co 
J 27 

0.313 -/00%. 



i.'^a 0.0/%. 



H i 




fn-^) 




\^ / 




£+ 




\ 


U7d 


A , 


f 


*-~ 




1.332 


0^- 


> 


' 



2.623- 
Z.286 



/.33 2 



0<.i/e ,^N.-|^ 



EXPLANATION OF PLATE XIV 

60 
A singles spectriim of Co showing 

the two photopeaks. The arrow shows 

the approximate setting of the 

discriminators. The extra peak to 

the right is a sxim peak due to the 

high counting rate used. 











70 


> 

X 










u 

1— 

< 

Q_ 








i 
\ 






• 




CHAMNEL 




^ 

N- 










) 


o 

u 
o 

-a 


1 

1 




' 




O — 

O 2 


i 
n 




1 


1 










giHOOO 












] 










"ii 



EXPLANATION OF PLATE XV 

A plot of the directional 
correlation data. 



73 



VIII. HEPEEENCES 



1. Adler, A., M. Palmai, and V. Perez-Hendez 

100 Mc/s tunnel diode discriminator and pulse 
shaper. Nuc. Instr. Meth. 15: 197. 1961. 

2. Artiges, H. C, and J. C. Brun 

Circuits a coincidences rapides utilisant des 
transistors a avalanche. J. Phys. Had. 22:53. 
1961. 

3. Bay, A. 

A new type of high-speed coincidence circuit. 
Eev. Sci. Instr. 22:397-^0. 1951. 

4. Bay, A. , and G. Papp « _q 

Coincidence device of 10" -lO"-^ second resolving 
power. Rev. Sci. Instr. 19:565' 19^. 

5. Beller, L. S. 

N-sec scintillation coincidence spectrometer 
system with high reliability. Hev. Sci. Instr. 
34:1001-1006. 1963. 

6. Biedenharn, L. C. , and M. E. Rose 

Theory of angular correlation of nuclear radiations. 
Phys. Rev. 25:729-777. 1953- 

7. Bjerke, Arthur A., Quentin A. Kerns, and Thomas A. 

Nunamaker. Pulse shaping and standardizing of photo- 
multiplier signals. Nuc. Instr. Meth. 15:249-267. 
1962. 

8. Bohan, W. A. , and A. J. Wager 

The effects of steady state and pulsed nuclear 
radiation on GaAs tunnel diodes. Ire Trans, on 
Nuc. Sci. NS-9(1): 346-354. 1962. 

9. Bonitz, M. 

Modern multi-channel time analyzers in the nano-. 
second range. Nuc. Instr. Meth. 22:238-252. 1963. 

10. Bret, George C, and Erwin P. Schrader 

Past tunnel diode circuits for Nal(Tl) detectors. 
Nuc. Instr. Meth. 13:177. 1961. 

11. Chase, Robert L. 

Multiplie coincidence circuit. Eev. Sci. Instr. 
31:945-949. I960. 



74 



12. Chase, RolDert L. ■ 

Nuclear Pulse Spectrometry. New York: McGraw-Hill, 
1961. 221 p. .. 

15. Dill, H. G. 

I;Tev; ways to trigger avalanche pulse circuits. 
Proc. IHS 49(6): 1095. 1961. 

14. Draegert, David A. 

l-Ieasuremen.ts of two properties of cascade gamma rays, 
Master's Thesis, Kansas State University, 1964. 

15. Dunworth, J. V. 

The application of the method of coincidence counting 
in nuclear physics. Rev. Sci. Instr. 11:167-180. 

1940. 

16. Elmore, V7. 

Coincidence circuit for a scintillation detector of 
radiation. Rev. Sci. Instr. 21:649. 1950. 

17. Peingold, A. M. , and Sherman Frankel 

Geometrical corrections in angular correlation 
measurements. Phys. Rev. 97:1025-1050. 1955. 

18. Prauenfelder, H. , and R. M. Steffen in 

Alpha-, beta-, gamma-ray spectroscopy. Amsterdam: 
North-Holland, 1965. 

19. Pusca, James A. 

Tunnel diodes may cut transistor costs. 
Aviation Week 71:7- 1959. 

20. Garwin, R. L. 

A useful fast coincidence circuit. Rev. Sci. Instr. 
21:569. 1950. 

21. Gentile, Sylvester P. 

Basic theory and application of tunnel diodes. 
Princeton: Van Nostrand, 1962. 295 pp. 

r 

22. Giorgis, J., and others. 

Tunnel diode manual. Liverpool, New York: General 
Electric, 1961. 97 pp. 

25. Gorodetsky, S. , and others. 

Circuit de coincidence a diodes t\innel. Nuc. Instr. 
Meth. 14:205-208. 1961. 



75 



24. Gorodetzky, S. , and others. 

Sur un circuit rapide de mise en forme a seuil 
reglable utilisant des diodes tunnel. Nuc. 
Instr. Meth. 13:282. 1961. 

25. Harrison, W. , and R. Poote. 

Tunnel diodes increase digital circuit switching 
speeds. Electronics 34(32) : 15-4-. 1961. 

26. Hazoni, Y. 

A fast flip-flop circuit utilizing tunnel diodes. 
Nuc. Instr. Meth. 13:95. 1961. 

27. Helmut, Paul 

Higher order chance coincidences in a fast-slow 
coincidence arrangement. Nuc. Instr. Meth. 9:131. 
I960. 

28. Hov7land, B. , and others. 

Electronics for Cosmic-Ray Experiments. 
Rev. Sci. Instr. 18:551. 19^7. 

29. Infante, G. , and F. Pandarese 

The tunnel diode as a threshold device; theory and 
application. Nuclear Electronics Vol. III. Vienna: 
International Atomic Energy Agency, 1962. 
pp 29-40. 

30. Kallmann, H. , and C. A. Accardo 

Coincidence experiments for noise reduction in 
scintillation counting. Rev. Sci. Instr. 21:48. 
1950. 

31. Krebs, A. T. 

Early history of the scintillation counter. Science 
122:17. 1955. 

32. Lawson, J. S. , and H. Prauenfelder 

The correction for finite angular resolution in 
directional correlation measurements. Phys. Rev. 
91:649-652. 1953. 

33. Lundby, A. 

Delayed coincidence circuit for scintillation 
counters. Rev. Sci. Instr. 22:324. 1951. 

34. Mayer-Kuckuk, T. , and R. Nierhaus 

Uber die bestimmung der zufalligen koinzidenzen 
in schnell-langsam koinzidenz-anordnungen. Nuc. 
Instr. Meth. 8:76-78. I960. 



76 



35 • McGervey, John D. 

A pulse resolver using tunnel diodes. Hue. Instr. 
Meth. 14:551-352. 1961. 

36. Miller, Barry 

Tunnel diode applications investigated. Aviation 
Week 71:72. 1959. 

37. Rose, M. E. 

The analysis of angular correlation and angular • i 

distribution data. Phys. Rev. 91:610-615- 1953. j 

i 

38. Schram, E. , and R. Lombaert ] 

Organic scintillation detectors. London: Elseview, I 
1963. I 

39* Shera, E. Brooks, K. J. Casper, and B. L. Robinson ] 

Analysis of chance coincidences in fast-slow coin- ^ 
cidence systems. Nuc. Instr. Meth. 24:482-492. 
1963. i 

40. Shera, E. Brooks 

Further comments on accidental coincidences in \ 

fast-slow coincidence systems. Uuc. Instr. Meth. ] 

12:198. 1961. 

41. Smaller, B. , and E. Avery • ) 

The use of gated-beam tubes in coincidence circuits. 

Rev. Sci. Instr. 22:341. 1951. ' 



42. Sprokel, G. J. 

A liquid scintillation counter using anticoincidence 
shielding. IBM Jour. Res. Dev. 7(2): 135-145. 1963. 

43. Strauss, M. G. 

Timing slow pulses for fast coincidence measurements. 
Rev. Sci. Instr. 34:1248. 1963- 

44. Tarnay, K. 

The maximxim power output of the tunnel diode 
oscillator. Proc. IRE 50(10) : 2120-2121. 1962. 

45. Van Zurk, R. 

Circuit discriminateur d' amplitude utilisant diodes 
tunnel. Nuc. Instr. Meth. 16:157-162. 1962. 

46. Ward, C. B. , and C. M. York 

A nanosecond pulse height discriminator. Nuc. Instr. 
Meth. 23:213-217. 1963. 



77 



4-7. Whetstone, Albert L. 

Improving the tunnel diode univibrator. Rev. Sci. 
Instr. 54:412-4-13. 1963- 

4-8. Whetstone, A., and S. Kounosu 

Nanosecond coincidence circuit using tunnel diodes. 
Rev. Sci. Instr. 33:4-23-4-28. 1962. 

4-9. Yonda, A. H. , R. Sugarman, and W. A. Higinbotham 

100 Mc counting system. Nuclear Electronics Vol. Ill, 
Vienna: International Atomic Energy Agency, 1962. 
pp. 3-13. 



A IffiASUREMENT OF THE Ni^*^ DIEECTIOITAL COREELATIOH 
WITH A TUMEL DIODE COINGIDEUCE. CIRCUIT , 



by 

GARY EDWIN GLAEK 
B. A., Park College, 1961 



AN ABSTRACT OF A MASTER'S THESIS 



submitted in partial fulfillmeat of the 



requirements for the degree 



MASTER OF SCIENCE 



Department of Physics 



KANSAS STATE UNIVERSITY 
Manhattan, Kansas. 



1966 



A coincidence circuit with a resolving time of a few 
nanoseconds has been built using tunnel diodes. The theory 
of operation is discussed. 

The circuit consists of a series of tunnel diode uni- 
vibrators. Germanium tunnel diodes are used at the inputs to 
give greater sensitivity and to reduce the time jitter. The 
first stage of each input is followed by a gallium arsenide 
tunnel diode univibrator. The purpose of this second stage 
is to provide a standard pulse of typically 5 usee width and 
100 mV height which is independent of the input pulse height. 

The standardized pulses are then fed to a coincidence 
univibrator through a germanium fast diode which serves to 
make the circuit unilateral. Standardized pulses from two 
separate circuits are added linearly in the coincidence unit. 
The coincidence t\innel diode univibrator is set to fire when 
the standardized pulses add to give a sum pulse of a certain 
height, which must be between one and two times the height of 
a singles standard pulse. The coincidence unit is followed 
by another xinivibrator which produces a much wider pulse. 
The output can then be counted by a scaler or used to gate a 
multichannel analyzer. 

A fast germanium diode is used to make the circuit uni- 
directional. With germanixim tunnel diodes in the circuit the 
voltage changes would be too small to effectively forward bias 
the fast diode, and the coupling would be primarily through 
the capacitance of the fast diode, which is bidirectional. 



For this reason gallium arsenide tunnel diodes, which, have a 
higher voltage change, are used. 

The directional correlation function of the well known 
Ni transitions was determined experimentally using the 
txinnel diode circuit. Corrections for detectors of finite 
size were made. The results agreed well with theory. 

Various experimental checks were made of coincidence 
rates, resolving times, and waveforms. All experimental 
results were found to agree well with theoretical predictions.